Optimal. Leaf size=186 \[ -\frac {a \left (2 a^2+3\right ) b^4 \tanh ^{-1}\left (\frac {1-a (a+b x)}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{8 \left (1-a^2\right )^{7/2}}-\frac {\left (11 a^2+4\right ) b^3 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^3 x}-\frac {5 a b^2 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^2 x^2}-\frac {b \sqrt {1-(a+b x)^2}}{12 \left (1-a^2\right ) x^3}-\frac {\sin ^{-1}(a+b x)}{4 x^4} \]
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Rubi [A] time = 0.27, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4805, 4743, 745, 835, 807, 725, 206} \[ -\frac {5 a b^2 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^2 x^2}-\frac {\left (11 a^2+4\right ) b^3 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^3 x}-\frac {a \left (2 a^2+3\right ) b^4 \tanh ^{-1}\left (\frac {1-a (a+b x)}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{8 \left (1-a^2\right )^{7/2}}-\frac {b \sqrt {1-(a+b x)^2}}{12 \left (1-a^2\right ) x^3}-\frac {\sin ^{-1}(a+b x)}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 206
Rule 725
Rule 745
Rule 807
Rule 835
Rule 4743
Rule 4805
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a+b x)}{x^5} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sin ^{-1}(x)}{\left (-\frac {a}{b}+\frac {x}{b}\right )^5} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\sin ^{-1}(a+b x)}{4 x^4}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\left (-\frac {a}{b}+\frac {x}{b}\right )^4 \sqrt {1-x^2}} \, dx,x,a+b x\right )\\ &=-\frac {b \sqrt {1-(a+b x)^2}}{12 \left (1-a^2\right ) x^3}-\frac {\sin ^{-1}(a+b x)}{4 x^4}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\frac {3 a}{b}+\frac {2 x}{b}}{\left (-\frac {a}{b}+\frac {x}{b}\right )^3 \sqrt {1-x^2}} \, dx,x,a+b x\right )}{12 \left (1-a^2\right )}\\ &=-\frac {b \sqrt {1-(a+b x)^2}}{12 \left (1-a^2\right ) x^3}-\frac {5 a b^2 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^2 x^2}-\frac {\sin ^{-1}(a+b x)}{4 x^4}-\frac {b^4 \operatorname {Subst}\left (\int \frac {-\frac {2 \left (2+3 a^2\right )}{b^2}-\frac {5 a x}{b^2}}{\left (-\frac {a}{b}+\frac {x}{b}\right )^2 \sqrt {1-x^2}} \, dx,x,a+b x\right )}{24 \left (1-a^2\right )^2}\\ &=-\frac {b \sqrt {1-(a+b x)^2}}{12 \left (1-a^2\right ) x^3}-\frac {5 a b^2 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^2 x^2}-\frac {\left (4+11 a^2\right ) b^3 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^3 x}-\frac {\sin ^{-1}(a+b x)}{4 x^4}+\frac {\left (a \left (3+2 a^2\right ) b^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\frac {a}{b}+\frac {x}{b}\right ) \sqrt {1-x^2}} \, dx,x,a+b x\right )}{8 \left (1-a^2\right )^3}\\ &=-\frac {b \sqrt {1-(a+b x)^2}}{12 \left (1-a^2\right ) x^3}-\frac {5 a b^2 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^2 x^2}-\frac {\left (4+11 a^2\right ) b^3 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^3 x}-\frac {\sin ^{-1}(a+b x)}{4 x^4}-\frac {\left (a \left (3+2 a^2\right ) b^3\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{b^2}-\frac {a^2}{b^2}-x^2} \, dx,x,\frac {\frac {1}{b}-\frac {a (a+b x)}{b}}{\sqrt {1-(a+b x)^2}}\right )}{8 \left (1-a^2\right )^3}\\ &=-\frac {b \sqrt {1-(a+b x)^2}}{12 \left (1-a^2\right ) x^3}-\frac {5 a b^2 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^2 x^2}-\frac {\left (4+11 a^2\right ) b^3 \sqrt {1-(a+b x)^2}}{24 \left (1-a^2\right )^3 x}-\frac {\sin ^{-1}(a+b x)}{4 x^4}-\frac {a \left (3+2 a^2\right ) b^4 \tanh ^{-1}\left (\frac {1-a (a+b x)}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{8 \left (1-a^2\right )^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 194, normalized size = 1.04 \[ \frac {1}{8} \left (\frac {a \left (2 a^2+3\right ) b^4 \log (x)}{\left (1-a^2\right )^{7/2}}-\frac {a \left (2 a^2+3\right ) b^4 \log \left (\sqrt {1-a^2} \sqrt {-a^2-2 a b x-b^2 x^2+1}-a^2-a b x+1\right )}{\left (1-a^2\right )^{7/2}}+\frac {b \sqrt {-a^2-2 a b x-b^2 x^2+1} \left (2 a^4-5 a^3 b x+a^2 \left (11 b^2 x^2-4\right )+5 a b x+4 b^2 x^2+2\right )}{3 \left (a^2-1\right )^3 x^3}-\frac {2 \sin ^{-1}(a+b x)}{x^4}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 484, normalized size = 2.60 \[ \left [-\frac {3 \, {\left (2 \, a^{3} + 3 \, a\right )} \sqrt {-a^{2} + 1} b^{4} x^{4} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) + 12 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} \arcsin \left (b x + a\right ) - 2 \, {\left ({\left (11 \, a^{4} - 7 \, a^{2} - 4\right )} b^{3} x^{3} - 5 \, {\left (a^{5} - 2 \, a^{3} + a\right )} b^{2} x^{2} + 2 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} b x\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{48 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} x^{4}}, -\frac {3 \, {\left (2 \, a^{3} + 3 \, a\right )} \sqrt {a^{2} - 1} b^{4} x^{4} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) + 6 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} \arcsin \left (b x + a\right ) - {\left ({\left (11 \, a^{4} - 7 \, a^{2} - 4\right )} b^{3} x^{3} - 5 \, {\left (a^{5} - 2 \, a^{3} + a\right )} b^{2} x^{2} + 2 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} b x\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{24 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.15, size = 1112, normalized size = 5.98 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 309, normalized size = 1.66 \[ -\frac {\arcsin \left (b x +a \right )}{4 x^{4}}-\frac {b \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{12 \left (-a^{2}+1\right ) x^{3}}-\frac {5 b^{2} a \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{24 \left (-a^{2}+1\right )^{2} x^{2}}-\frac {5 b^{3} a^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{8 \left (-a^{2}+1\right )^{3} x}-\frac {5 b^{4} a^{3} \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{8 \left (-a^{2}+1\right )^{\frac {7}{2}}}-\frac {3 b^{4} a \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{8 \left (-a^{2}+1\right )^{\frac {5}{2}}}-\frac {b^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{6 \left (-a^{2}+1\right )^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {asin}\left (a+b\,x\right )}{x^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asin}{\left (a + b x \right )}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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